Introduction to System Dynamics

Abstract

System Dynamics is the study of the change of the power variables within an energetic system. As a system interacts with a power source, energy flows across the system boundary, between storage modes within the system, and is dissipated as heat by friction, electrical resistance, or magnetic hysteresis. Modeling energetic systems as circuit-like networks simplifies the accounting of power flows in the system. Creation of a mathematical model of an energetic system represented as a network is straightforward elimination by substitution, guided by a drawing of the network, once a complete set of mathematical statements of the physical truths of network-like system has been written.

Problems

Problem 1.1 Translational mechanical power is the dot product of force and the velocity of the point of application of the force, \(\mathsf{\mathbb{P}}=\mathbf{F}\cdot \mathbf{v}\) When the force and velocity of the point of application of the force are colinear, the scalar equation \(\mathsf{\mathbb{P}}=F\,v\) can be used. A mechanical system was de-energized before the force shown in Fig. P1.1a with the colinear velocity shown in Fig. P1.1b acted on it.

1.1.b What is the minimum horsepower motor which could provide the power needed?

Problem 1.2 DC Electrical power is product of current times voltage, \(\mathsf{\mathbb{P}}=i\,v\) A DC motor and the system it drives was de-energized, before the DC motor’s power source provided it the current i at the voltage v, shown in Fig. P.1.2a and b.

1.2.b What is the minimum horsepower motor which could accept the power from source?

Problem 1.3 A piping system is shown in Fig. P1.3. The fluid is modeled as incompressible. Consequently, the continuity equations can be written in terms of volume flow rate, Q, rather than in terms of mass flow rate. The branches are identified by letter, and pressure nodes between the branches are numbered.

Piping system schematic. The fluid is modeled as incompressible. There is fluid resistance in each branch, which decreases pressure in the direction of the fluid flow

1.3.a Orient the flow in each branch. The positive direction for flow through pump is from node 5 to node 1.

1.3.b Write a complete set of independent compatibility equations in the form of path equations.

1.3.c Write a complete set of independent continuity equations for volume flow rate.

Problem 1.4 A piping system is shown in Fig. P1.4. The fluid is modeled as incompressible. Consequently, the continuity equations can be written in terms of volume flow rate, Q, rather than in terms of mass flow rate. The branches are identified by letter, and pressure nodes between the branches are numbered.

1.6.a Calculate the energy transferred to the specimen for the displacement, \( x=4\,\text{in,} \) in foot-pounds and joules.

1.6.b Calculate the coenergy for the displacement, \( x=4\,\text{in}\text{.} \)

Problem 1.7 An electric circuit consisting of a voltage source, represented as a battery, a switch, a resistor, an inductor, and a capacitor, is shown in Fig. P1.7. The energetic equations of this circuit are listed. Use elimination by substitution to derive the system equation for the input voltage and output variable indicated:

Problem 1.8 An electric circuit consisting of a voltage source, a resistor, an inductor, and a capacitor, annotated with nodes of distinct values of voltage and arrows indicating the positive direction of current through each element, is shown in Fig. P1.8. The energetic equations of this circuit are listed. Use elimination by substitution to derive the system equation for the input and output variable indicated:

Problem 1.9 An electric circuit consisting of a voltage source, a resistor, an inductor, and a capacitor is shown in the schematic Fig. P1.9. The energetic equations of this circuit are listed. Use elimination by substitution to derive the system equation for the input and output variable indicated:

Problem 1.10 A translational mechanical system consisting of a mass M sliding on a lubricating fluid film with damping b, and a spring K attached between the mass and ground is shown in Fig. P1.10a. The linear graph of this energetic system, analogous to an electric circuit diagram is Fig. P1.10b. The energetic equations are listed. Use elimination by substitution to derive the system equation for the input and output variable indicated:

Problem 1.11 A schematic of a hydraulic system is shown in Fig. P1.11a. The pump, modeled as a pressure source p(t), discharges fluid into a hydraulic circuit consisting of two fluid resistances, R1 and R2, a fluid inertance I, which stores kinetic energy, and a fluid accumulator with capacitance C, which stores energy by compressing a spring or nitrogen-filled bladder. Figure P1.11b is the linear graph of the system, analogous to an electric circuit. The energetic equations are listed. Use elimination by substitution to derive the system equation for the input and output variable indicated: